Some help rendering the Mandelbrot set

Question

I have been given some work to do with the fractal visualisation of the Mandelbrot set.

I'm not looking for a complete solution (naturally), I'm asking for help with regard to the orbits of complex numbers.

Say I have a given Complex number derived from a point on the complex plane. I now need to iterate over its orbit sequence and plot points according to whether the orbits increase by orders of magnitude or not.

How do I gather the orbits of a complex number? Any guidance is much appreciated (links etc). Any pointers on Math functions needed to test the orbit sequence e.g. Math.pow()

I'm using Java but that's not particularly relevant here.

Thanks again, Alex

Answer 1

When you display the Mandelbrot set, you simply translate the real and imaginaty planes into x and y coordinates, respectively.

So, for example the complex number 4.5 + 0.27i translates into x = 4.5, y = 0.27.

The Mandelbrot set is all points where the equation Z = Z² + C never reaches a value where |Z| >= 2, but in practice you include all points where the value doesn't exceed 2 within a specific number of iterations, for example 1000. To get the colorful renderings that you usually see of the set, you assign different colors to points outside the set depending on how fast they reach the limit.

As it's complex numbers, the equation is actually Zr + Zi = (Zr + Zi)² + Cr + Ci. You would divide that into two equations, one for the real plane and one for the imaginary plane, and then it's just plain algebra. C is the coordinate of the point that you want to test, and the initial value of Z is zero.

Here's an image from my multi-threaded Mandelbrot generator :)

Answer 2

Actually the Mandelbrot set is the set of complex numbers for which the iteration converges.

So the only points in the Mandelbrot set are that big boring colour in the middle. and all of the pretty colours you see are doing nothing more than representing the rate at which points near the boundary (but the wrong side) spin off to infinity.

In mathspeak,

M = {c in C : lim (k -> inf) z_k = 0 } where z_0 = c, z_(k+1) = z_k^2 + c

ie pick any complex number c. Now to determine whether it is in the set, repeatedly iterate it z_0 = c, z_(k+1) = z_k^2 + c, and z_k will approach either zero or infinity. If its limit (as k tends to infinity) is zero, then it is in. Otherwise not.

It is possible to prove that once |z_k| > 2, it is not going to converge. This is a good exercise in optimisation: IIRC |Z_k|^2 > 2 is sufficient... either way, squaring up will save you the expensive sqrt() function.

Answer 3

Wolfram Mathworld has a nice site talking about the Mandelbrot set.

A Complex class will be most helpful.

Maybe an example like this will stimulate some thought. I wouldn't recommend using an Applet.

You have to know how to do add, subtract, multiply, divide, and power operations with complex numbers, in addition to functions like sine, cosine, exponential, etc. If you don't know those, I'd start there.

The book that I was taught from was Ruel V. Churchill "Complex Variables".

Answer 4

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